Note: I have references for many of these things, but Friday morning stream-of-consciousness at the moment. References later if I get a chance.
There are a lot of different issues here, though all generally related in the "How should/could/does Halachah change over time?" sense.
This is perhaps the classic one. They actually had a technological solution at the time - signal fires, though that had issues with making sure the fires were legitimate. Some things really never change - it is still trivially easy to have a news report, Tweet, etc. go around the world and find out later (whether 5 minutes or a day) that it was totally false. Sometimes due to malicious intent, sometimes accidental.
That being said, the classic response for Yom Tov Sheni is "Minhag avoseinu beyadeinu" - essentially, we do what was done before because it was done before. In my opinion, that's the only reason that validates why we have Yom Tov Sheni for Shavuos when (a) it is 7 weeks after Pesach and (b) we don't count the Omer "Maybe today is the n'th day, maybe it is n+1'th day." As far as the Sanhedrin, my understanding is that it is because we can't go back to "witnesses to declare Rosh Chodesh" without the Sanhedrin, so therefore we also can't go to a new mode of calculating the calendar since that was established to take the temporary place of the Sanhedrin & witnesses. Note also that the current calendar is based on average Molad, which is incredibly accurate long-term but often quite a bit off in a particular month, and because of a very rigid set of rules based around the days for Yom Kippur (can't have it on Friday or Sunday) and Hoshanah Rabbah (can't have it on Shabbos) and other factors, we sometimes end up with days for Rosh Chodesh that are, based on pure astronomical information, a day off.
We start saying Mashiv Haruach Umorid Hagashem on Shemini Atzeres. But the date for saying V'ten Tal Umatar varies. In particular, we typically say it outside Israel starting around December 5. However, it is supposed to be 60 days after Tekufas Tishrei - which is the autumnal equinox. There are two separate issues:
1 - This timing makes sense for Bavel (current Iraq). It doesn't make sense in many other parts of the world where either the seasons are flipped (southern hemisphere) or the rainy season is simply different. There actually is a Halachic basis for saying V'ten Tal Umatar at different times in some locations, but it gets complicated.
2 - I posit that this really should be astronomically based. The problem is the Julian calendar (365.25 days per year) vs. the Gregorian calendar (365.2425 days per year). The change was the subject of much debate in both religious and secular circles. The date for V'ten Tal Umatar is based on a pure 365.25 day year, with no adjustment. As a result, while Pesach stays in the spring (as it must), V'ten Tal Umatar has been creeping later, roughly 3 days per 400 years. I have discussed this with many Rabbis, and most of them either have no idea what I'm talking about (I am a bit of time/date geek) or have an answer along the lines of "the Rabbis knew that 365.25 wasn't perfect but they stuck with it because anything else would have been too complicated".
IMHO, this is a situation where it should be changed, but would likely need the Sanhedrin to do so, as it isn't simply "correct the Halachah based on new information" but rather "change our decision in light of the ability to use the information".
When it comes to material things, there seems to be a lot more flexibility to make use of new items - whether totally new (plastic) or simply unknown (birds & animals).
By this I would, broadly (and possibly incorrectly) include: materials for specific uses such as the pomegranate branch for the Korban Pesach (where such materials are not specifically mandated by the Torah - e.g., the classic example being Ketores (incense) where thousands of years ago they knew of things that might be "good" for Ketores but which were not added because there was a specific formula in the Torah), which animals/birds are Kosher (e.g., the issue of turkey, previously unknown types of fish, etc.), Kashering and Toveling rules for plastic and other new materials (e.g., plastic does not need Toveling because it was not in the specific categories where the Torah requires it; the Kashering process for plastic varies from "trivial" to "impossible", depending on the type of plastic, the type of use, how well it can be cleaned, etc.).
One specific example that come to mind include Sukkah - we can't make Schach itself out of plastic, but according to many opinions you can use certain plastic (monofilament) to tie together Schach. Similarly, many people use a rubber band (unknown thousands of years ago) to hold together the 5 Hoshanos.
That still doesn't answer the question of using a carbon composite rod as the spit for a Korban Pesach. The pomegranate growers union would be against it, but arguably if there is no fundamental but unstated reason for it then it would seem to me (but I am not a Rabbi) that this might be perfectly fine, provided the new material was shown to have similar properties (thermal conductivity, moisture, etc.) to the original object, but only because this is a secondary object (like the plastic with Schach or rubber with Hoshanos) and not the primary object (which has very specific requirements - type of animal, age, etc.)